\(KK\)-theory of A-valued semi-circular systems
Let \(E\) be a Hilbert \(A\)-module with a \(^*\)-homomorphism from \(A\) to the \(C^*\)-algebra of adjointable operators on \(E\). A representation of \(E\) on a \(C^*\)-algebra \(B\) is a pair consisting of a \(^*\)-homomorphism \(f\) from \(A\) to and a linear map \(S\) from \(E\) to \(B\) with \(S(a_1 x a_2) = f(a_1) S(x) f(a_2)\) for all \(a_1,a_2\in A\), \(x\in E\) and \(S(x_1)^* S(x_2) = f(\langle x_1,x_2\rangle)\) for all \(x_1,x_2\in E\). The Toeplitz \(C^*\)-algebra \(T_E\) is the universal \(C^*\)-algebra for such representations.NEWLINENEWLINE\textit{M. V. Pimsner} [Fields Inst. Commun. 12, 189--212 (1997; Zbl 0871.46028)] has shown that the canonical inclusion of \(A\) into \(T_E\) is a \(KK\)-equivalence. Here this is generalised as follows: the inclusion of \(A\) into \(S\) is a \(KK\)-equivalence, where \(S\) is a \(C^*\)-subalgebra of \(T_E\) generated by \(A\) and some set of linear combinations of creation and annihilation operators. In particular, if \(E\) carries an involution, \(S\) could be the \(C^*\)-subalgebra generated by \(A\) and elements of the form \(T_x + T_{x^*}^*\) for \(x\in E\), where \(T_x\) denotes the element of \(T_E\) associated to \(x\in E\). This contains a \(K\)-theory computation by \textit{D. Voiculescu} [K-Theory 7, No. 1, 5--7 (1993; Zbl 0786.46054)] for a semicircular family as a special case.
scientific article; zbMATH DE number 6579852